Skip to main content
SpringerLink
Log in

The functional equation of multiplicative derivation is superstable on standard operator algebras

  • Short Communications
  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

LetX be a real or complex infinite dimensional Banach space andA a standard operator algebra onX. Denote byB(X) the algebra of all bounded linear operators onX. Let ϕ: ℝ+ → ℝ+ be a function with the property lim t→∞ φ(t)t −1=0. Assume that a mappingD:A →B(X) satisfies ‖D(AB)−AD(B)−D(A)B‖<φ(‖A‖ ‖B‖) for all operatorsA, B ∈D (no linearity or continuity ofD is assumed). ThenD is of the formD(A)=AT−TA for someT∈B(X).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. J.A. Baker, The stability of the cosine equation,Proc. Amer. Math. Soc. 80 (1980), 411–416.

    Google Scholar 

  2. D.G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings,Duke Math. J. 16 (1949), 385–397.

    Google Scholar 

  3. P.R. Chernoff, Representations, automorphisms and derivations of some operator algebras,J. Funct. Anal. 12 (1973), 275–289.

    Google Scholar 

  4. M.N. Daif, When is a multiplicative derivation additive?,Internat. J. Math. & Math. Sci. 14 (1991), 615–618.

    Google Scholar 

  5. D.H. Hyers, On the stability of the linear functional equation,Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224.

    Google Scholar 

  6. D.H. Hyers, The stability of homomorphisms and related topics, in: Global Analysis-Analysis on manifolds Th.M. Rassias, ed., Teubner-Texte zur Mathematik, Leipzig (1983), 140–153.

    Google Scholar 

  7. D.H. Hyers and Th.M. Rassias, Approximate homomorphisms, to appear.

  8. P. Šemrl, Additive derivations of some operator algebras,Illinois J. Math. 35 (1991), 234–240.

    Google Scholar 

  9. P. Šemrl, Ring derivations on standard operator algebras, to appear inJ. Funct. Anal.

  10. S.M. Ulam, A collection of mathematical problems, Interscience, New York, 1960.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Ljubljana, Jadranska 19, 61000, Ljubljana, Slovenia

    Peter Šemrl

Authors
  1. Peter Šemrl
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was supported by the Research Council of Slovenia

Rights and permissions

Reprints and permissions

About this article

Cite this article

Šemrl, P. The functional equation of multiplicative derivation is superstable on standard operator algebras. Integr equ oper theory 18, 118–122 (1994). https://doi.org/10.1007/BF01225216

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01225216

Keywords

Search

Navigation